Brownian Motor

In his famous lectures Richard Feynman discussed the impossibility to violate the second law of thermodynamics by a ratchet mechanism. The simplest model for a ratchet is an overdamped Brownian particle in an asymmetric but spatially periodic potential (with asymmetry and period L). Due to the fluctuating force caused by the pushing molecules of the surrounding fluid or gas the Brownian particle may overcome the potential barrier moving to the left or to the right. The probabilities for both directions are equal. Thus on average the particle does not move. Hence building a motor which turns thermal energy into mechanical work from a single heat bath is impossible.

But the ratchet can be turned into a so-called a Brownian motor that seems to violate the second law of thermodynamics. The idea is to turn the ratchet potential periodically on and off with a frequency 1/2. Under certain circumstances this may yield directed motion even against an applied force f. It is indeed a device doing work. Below you can play with a Java applet which simulates such a system. Vary the parameters and figure out the condition for a drift from right to left even for f > 0. If you want to know why a Brownian motor isn't a perpetuum mobile of the second kind click on here.

Why is a Brownian motor not a perpetuum mobile of the second kind?

As long as the ratchet potential is off the particle will move diffusively according to a (biased) random walk, leading to a variance in position of and a mean position of , where D = kT / is the diffusion constant. When the ratchet potential is switched on, the particle gets trapped in one of the potential minima. If for the variance holds, the particle on average gets trapped into the minimum left to the starting point. The maximum flux is obtained if the switching time is large enough to assure that the particle can adjust in the trapping minimum ('adiabatic adjustment time') and also is small enough to fulfill the above requirement for the variance. One can say roughly that a net flux to the left always occurs, when thermal energy is significantly smaller than the potential maximum, the external force is chosen not too big and the driving frequency matches the adiabatic adjustment time needed for the particle to move into a potential minima.

Where does the energy come from leading to a drift against the external force? The energy does not come from the heat bath but from the ratchet potential when it is switched on. At that moment the potential energy of the particle will be suddenly increased. In the simulation this can be seen by a sudden increase of the energy bar. But most of the energy pushed into the system will be just dissipated into the heat bath due to the relaxation of the particle into a potential minima. Only a tiny portion will be used for doing work. Thus a Brownian motor does not violate any law of thermodynamics it only turns one type of work into another one. Nevertheless the fluctuating force due to the heat bath is essential for a Brownian motor.

For more details and possible applications in biology and chemistry read the following review article:
R.D. Astumian: Thermodynamics and Kinetics of a Brownian Motor, Science 276, p. 917-922 (1997).